Abstract
The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an infinite regular graph. We relate quantitatively the empirical measure of the eigenvalues and the delocalization of the eigenvectors to the spectrum of the adjacency operator of the percolation on the infinite graph. Secondly, we prove that percolation on an infinite regular tree with degree at least three preserves the existence of an absolutely continuous spectrum if the removal probability is small enough. These two results are notably relevant for bond percolation on a uniformly sampled regular graph or a Cayley graph with large girth.
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Aizenman M., Sims R., Warzel S.: Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs. Probab. Theory Relat. Fields 136(3), 363–394 (2006)
Aizenman M., Warzel S.: Resonant delocalization for random Schrödinger operators on tree graphs. J. Eur. Math. Soc. (JEMS) 15(4), 1167–1222 (2013)
Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab. 12(54):1454–1508 (electronic) (2007)
Anantharaman, N., Le Masson, E.: Quantum ergodicity on large regular graphs. arXiv:1304.4343 [math-ph]
Anderson P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)
Antunović, T., Veselić, I.: Spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs. In: Methods of spectral analysis in mathematical physics, volume 186 of Oper. Theory Adv. Appl. Birkhäuser, Basel, pp. 1–29 (2009)
Athreya, K.B., Ney, P.E.: Branching processes. Springer, New York, Die Grundlehren der mathematischen Wissenschaften, Band 196 (1972)
Bordenave, C.: Spectral measures of random graphs. (2014). http://www.math.univ-toulouse.fr/bordenave/
Bordenave C., Caputo P., Chafaï D.: Spectrum of non-Hermitian heavy tailed random matrices. Comm. Math. Phys. 307(2), 513–560 (2011)
Bordenave C., Guionnet A.: Localization and delocalization of eigenvectors for heavy-tailed random matrices. Probab. Theory Relat. Fields 157(3-4), 885–953 (2013)
Bordenave C., Lelarge M.: Resolvent of large random graphs. Random Struct. Algorithms 37(3), 332–352 (2010)
Bordenave C., Lelarge M., Salez J.: The rank of diluted random graphs. Ann. Probab. 39(3), 1097–1121 (2011)
Bordenave, C., Sen, A., Virág, B.: Mean quantum percolation. arXiv:1308.3755
Brooks S., Lindenstrauss E.: Non-localization of eigenfunctions on large regular graphs. Israel J. Math. 193(1), 1–14 (2013)
Chayes J.T., Chayes L., Franz J.R., Sethna J.P., Trugman S.A.: On the density of states for the quantum percolation problem. J. Phys. A 19(18), L1173–L1177 (1986)
de Gennes P.G., Lafore P., Millot J.P.: Amas accidentels dans les solutions solides désordonnées. J. Phys. Chem. Solids 11(12), 105–110 (1959)
de Gennes P.G., Lafore P., Millot J.P.: Sur un phénomène de propagation dans un milieu désordonné. J. Phys. Rad. 20, 624 (1959)
DeVore, R.A., Lorentz, G.G.: Constructive approximation. Volume 303 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1993)
Dumitriu I., Pal S.: Sparse regular random graphs: spectral density and eigenvectors. Ann. Probab. 40(5), 2197–2235 (2012)
Duren P.L.: Theory of H p spaces. Pure and applied mathematics, Vol. 38. Academic Press, New York (1970)
Durrett R.: Random graph dynamics. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge (2010)
Erdős, L., Schlein, B., Yau, H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. IMRN (3), 436–479 (2010)
Flajolet P., Sedgewick R.: Analytic combinatorics. Cambridge University Press, Cambridge (2009)
Froese R., Halasan F., Hasler D.: Absolutely continuous spectrum for the Anderson model on a product of a tree with a finite graph. J. Funct. Anal. 262(3), 1011–1042 (2012)
Froese R., Hasler D., Spitzer W.: Absolutely continuous spectrum for the Anderson model on a tree: a geometric proof of Klein’s theorem. Comm. Math. Phys. 269(1), 239–257 (2007)
Froese, R., Hasler, D., Spitzer, W.: A geometric approach to absolutely continuous spectrum for discrete Schrödinger operators. In: Random walks, boundaries and spectra. Volume 64 of Progr. Probab., pp. 201–226. Birkhäuser/Springer, Basel (2011)
Geisinger, L.: Convergence of the density of states and delocalization of eigenvectors on random regular graphs. arXiv:1305.1039
Golénia S., Schumacher C.: The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs. J. Math. Phys. 52(6), 063512, 17 (2011)
Halasan F.: Absolutely continuous spectrum for the Anderson model on some tree-like graphs. Ann. Henri Poincaré 13(4), 789–811 (2012)
Kallenberg O.: Foundations of modern probability. Probability and its applications (New York) second edition. Springer, New York (2002)
Keller M.: Absolutely continuous spectrum for multi-type Galton Watson trees. Ann. Henri Poincaré 13(8), 1745–1766 (2012)
Keller M., Lenz D., Warzel S.: Absolutely continuous spectrum for random operators on trees of finite cone type. J. Anal. Math. 118(1), 363–396 (2012)
Kirkpatrick S., Eggarter T.P.: Localized states of a binary alloy. Phys. Rev. B 6, 3598–3609 (1972)
Klein A.: Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133(1), 163–184 (1998)
Klein A., Sadel C.: Ballistic behavior for random Schrödinger operators on the Bethe strip. J. Spectr. Theory 1(4), 409–442 (2011)
Klein A., Sadel C.: Absolutely continuous spectrum for random Schrödinger operators on the Bethe strip. Math. Nachr. 285(1), 5–26 (2012)
Kottos T., Smilansky U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Phys. 274(1), 76–124 (1999)
Li W.-C.W, Patrick Solé P.: Spectra of regular graphs and hypergraphs and orthogonal polynomials. Eur. J. Combin. 17(5), 461–477 (1996)
McKay, B.D., Wormald, N.C., Wysocka, B.: Short cycles in random regular graphs. Electron. J. Combin. 11 (1), Research Paper pp. 66, 12 (electronic) (2004)
Müller, P., Stollmann, P.: Percolation Hamiltonians. In: Random walks, boundaries and spectra. Volume 64 of Progr. Probab., pp. 235–258. Birkhäuser/Springer, Basel (2011)
Schumacher, C., Schwarzenberger, F.: Approximation of the integrated density of states on sofic groups. Annales Henri Poincaré, pp. 1–35 (2014)
Shirai, T.: Limit theorems for random analytic functions and their zeros. In: Functions in number theory and their probabilistic aspects, RIMS Kôkyûroku Bessatsu, B34, pages 335–359. Res. Inst. Math. Sci. (RIMS), Kyoto (2012)
Simon B.: Trace ideals and their applications. Volume 120 of mathematical surveys and monographs second edition. American Mathematical Society, Providence, RI (2005)
Tao T., Vu V.: Random matrices: universality of local eigenvalue statistics. Acta Math. 206(1), 127–204 (2011)
Terras A.: Finite quantum chaos. Am. Math. Monthly 109(2), 121–139 (2002)
Terras, A.: Zeta functions of graphs. Volume 128 of Cambridge Studies in Advanced Mathematics. A stroll through the garden. Cambridge University Press, Cambridge (2011)
Tran L.V., Vu V.H., Wang Ke.: Sparse random graphs: eigenvalues and eigenvectors. Random Struct. Algorithms 42(1), 110–134 (2013)
Veselić I.: Spectral analysis of percolation Hamiltonians. Math. Ann. 331(4), 841–865 (2005)
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Communicated by Anton Bovier.
Research was partially supported by ANR-11-JS02-005-01.
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Bordenave, C. On Quantum Percolation in Finite Regular Graphs. Ann. Henri Poincaré 16, 2465–2497 (2015). https://doi.org/10.1007/s00023-014-0382-9
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DOI: https://doi.org/10.1007/s00023-014-0382-9